Abstract: Let be a locally compact topological group and be, respectively, the Fourier and Fourier-Stieltjes algebras of . It is one of the purposes of this paper to investigate the (= Radon-Nikodym property) and some other geometric properties such as weak , the Dunford-Pettis property and the Schur property on the algebras and , and to relate these properties to the properties of the multiplication operator on the group -algebra . We also investigate the problem of Arens regularity of the projective tensor products , when has the and is any -algebra. Some related problems on the measure algebra, the group algebra and the algebras , , are also discussed.